Volume 68, Issue 2

We prove some results about additive classes of functions with Lebesgue null or meager closures. We consider also families with closures in the Mendez ideals.

Originally, partial information systems were introduced as a means of providing a representation of the Smyth powerdomain in terms of order convex substructures of an information-based structure. For every partial information system 𝕊, there is a new partial information system that is natrually induced by the principal lowersets of the consistency predicate for 𝕊. In this paper, we show that this new system serves as a completion of the parent system 𝕊 in two ways. First, we demonstrate that the induced system relates to the parent system 𝕊 in much the same way as the ideal completion of the consistency predicate for 𝕊 relates to the consistency predicate itself. Second, we explore the relationship between this induced system and the notion of D -completions for posets. In particular, we show that this induced system has a “semi-universal” property in the category of partial information systems coupled with the preorder analog of Scott-continuous maps that is induced by the universal property of the D -completion of the principal lowersets of the consistency predicate for the parent system 𝕊.

An ideal I of a ring A is a z -ideal if whenever a , b ∈ A belong to the same maximal ideals of A and a ∈ I , then b ∈ I as well. On the other hand, an ideal J of A is a d -ideal if Ann 2 ( a ) ⊆ J for every a ∈ J . It is known that the lattices Z( L ) and D( L ) of the ring 𝓡 L of continuous real-valued functions on a frame L , consisting of z -ideals and d -ideals of 𝓡 L , respectively, are coherent frames. In this paper we characterize, in terms of the frame-theoretic properties of L (and, in some cases, the algebraic properties of the ring 𝓡 L ), those L for which Z( L ) and D( L ) satisfy the various regularity conditions on algebraic frames introduced by Martínez and Zenk [20]. Every frame homomorphism h : L → M induces a coherent map Z( h ) : Z( L ) → Z( M ). Conditions are given of when this map is closed, or weakly closed in the sense Martínez [19]. The case of openness of this map was discussed in [11]. We also prove that, as in the case of the ring C( X ), the sum of two z -ideals of 𝓡 L is a z -ideal.

Partial frames provide a fertile context in which to do pointfree structured and unstructured topology, using a small collection of axioms of an elementary nature. Amongst other things they can be used to investigate similarities and differences between frames, σ -frames and κ -frames. In this paper, the theory of strong inclusions for partial frames is used to describe compactifications of completely regular partial frames; the elements of these compactifications are given explicitly as strongly regular ideals. This is independent of and encompasses the theory of compactifications for frames. As an application, we revisit the Samuel compactification of a uniform partial frame.

Let G be an abelian ℓ -group with a strong order unit u > 0. We call G u -clean after Hager, Kimber, and McGovern if every element of G can be written as a sum of a strong order unit of G and a u -component of G . We prove that G is u -clean if and only if u -components of G can be lifted modulo any ℓ -ideal of G . Moreover, we introduce a notion of u -suitable ℓ -groups (as a natural analogue of the corresponding notion in Ring Theory) and we prove that the ℓ -group G is u -clean when and only when it is u -suitable. Also, we show that if E is a vector lattice, then E is u -clean if and only if the space of all u -step functions of E is u -uniformly dense in E . As applications, we will generalize a result by Banaschewski on maximal ℓ -ideals of an archimedean bounded f -algebras to the non-archimedean case. We also extend a result by Miers on polynomially ideal C ( X )-type algebras to the more general setting of bounded f -algebras.

The present paper deals with invariance of nonatomic measures defined on effect algebras. Firstly, it is proved that if μ is a nonatomic and continuous probability measure defined on a σ -complete effect algebra L , then it satisfies para-Darboux property. Then, the invariance between continuous probability measures m and μ defined on a σ -complete effect algebra L is established when μ is nonatomic satisfying para-Darboux property on L .

We consider the Alexander dual of path ideals of cycle posets, and we compute the Castelnuovo-Mumford regularity. As a consequence, we get the projective dimension of path ideals of cycle posets. Our results are expressed in terms of the combinatorics of the underlying poset.

In the present paper, we introduce the notion of generalized multiplicative derivation in a near-ring N and investigate commutativity of 3-prime near-rings, showing that certain conditions involving generalized multiplicative derivations force N to be a commutative ring. Finally some more results related with the structure of these derivations are also obtained.

In this paper we introduce the notions of quasi-entwining structure and cleft extension for a quasi-entwining structure. We prove that if ( A , C , ψ ) is a quasi-entwining structure and the associated extension to the submagma of coinvariants A C is cleft, there exists an isomorphism ω A between A C ⊗ C and A . Moreover, we define two unital but not necessarily associative products on A C ⊗ C . For these structures we obtain the necessary and sufficient conditions to assure that ω A is a magma isomorphism, giving some examples fulfilling these conditions.

We show that given a finitely generated LERF group G with positive rank gradient, and finitely generated subgroups A , B ≤ G of infinite index, one can find a finite index subgroup B 0 of B such that [ G : 〈 A ∪ B 0 〉] = ∞. This generalizes a theorem of Olshanskii on free groups. We conclude that a finite product of finitely generated subgroups of infinite index does not cover G . We construct a transitive virtually faithful action of G such that the orbits of finitely generated subgroups of infinite index are finite. Some of the results extend to profinite groups with positive rank gradient.

The error function occurs widely in multiple areas of mathematics, mathematical physics and natural sciences. There has been no work in this area for the past four decades. In this article, we estimate the coefficient bounds with q-difference operator for certain classes of the spirallike starlike and convex error function associated with convolution product using subordination as well as quasi-subordination. Though this concept is an untrodden path in the field of complex function theory, it will prove to be an encouraging future study for researchers on error function.

In this paper, we use the Faber polynomial expansion to find upper bounds for | a n | ( n ≥ 3) coefficients of functions belong to classes HqΣ(λ,h),STqΣ(α,h) andMqΣ(α,h)$\begin{array}{} H_{q}^{\Sigma}(\lambda,h),\, ST_{q}^{\Sigma}(\alpha,h)\,\text{ and} \,\,M_{q}^{\Sigma}(\alpha,h) \end{array}$ which are defined by quasi-subordinations in the open unit disk 𝕌. Further, we generalize some of the previously published results.

In this paper, we establish new results on the existence of positive periodic solutions for the following high-order neutral functional differential equation (NFDE) (x(t)−cx(t−σ))(2m)+f(x(t))x′(t)+g(t,x(t−δ))=e(t).$$\begin{array}{} (x(t)-cx(t-\sigma)) ^{(2m)}+f(x(t)) x'(t)+g(t,x(t-\delta))=e(t). \end{array}$$ The interesting thing is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the corresponding ones known in the literature. Two examples are given to illustrate the effectiveness of our results.

We obtain in terms of additive and multi-additive functions the solutions f , h : S → H of the functional equation ∑λ∈Φf(x+λy+aλ)=Nf(x)+h(y),x,y∈S,$$\begin{array}{} \displaystyle \sum\limits_{\lambda \in \Phi }f(x+\lambda y+a_{\lambda })=Nf(x)+h(y),\quad x,y\in S, \end{array} $$ where ( S , +) is an abelian monoid, Φ is a finite group of automorphisms of S , N = | Φ | designates the number of its elements, { a λ , λ ∈ Φ} are arbitrary elements of S and ( H , +) is an abelian group. In addition, some applications are given. This equation provides a joint generalization of many functional equations such as Cauchy’s, Jensen’s, Łukasik’s, quadratic or Φ-quadratic equations.

If X and Y are FK spaces, then every infinite matrix A ∈ ( X , Y ) defines a bounded linear operator L A ∈ B ( X , Y ) where L A ( x ) = Ax for each x ∈ X . But the converse is not always true. Indeed, if L is a general bounded linear operator from X to Y , that is, L ∈ B ( X , Y ), we are interested in the representation of such an operator using some infinite matrices. In this paper we establish the representations of the general bounded linear operators from the space bv into the spaces ℓ ∞ , c and c 0 . We also prove some estimates for their Hausdorff measures of noncompactness. In this way we show the difference between general bounded linear operators between some sequence spaces and the matrix operators associated with matrix transformations.

In contrast to the subspaces of all C -symmetric operators, we show that the subspaces of all skew-C symmetric operators are reflexive and even hyperreflexive with the constant κ (𝓒 s )≤ 3.

In our paper we will continue the comparison which was started by Vsevolod I. Ivanov [Nonlinear Analysis 125 (2015), 270–289], where he compared scalar optimality conditions stated in terms of Hadamard derivatives for arbitrary functions and those which was stated for ℓ -stable functions in terms of Dini derivatives. We will study the vector optimization problem and we show that also in this case the optimality condition stated in terms of Hadamard derivatives is more advantageous.

The symbol 𝓢 c ( X ) denotes the hyperspace of all nontrivial convergent sequences in a Hausdorff space X . This hyperspace is endowed with the Vietoris topology. In the current paper, we compare the cellularity, the tightness, the extent, the dispersion character, the net weight, the i-weight, the π -weight, the π -character, the pseudocharacter and the Lindelöf number of 𝓢 c ( X ) with the corresponding cardinal function of X . We also answer a question posed by the authors in a previous paper.

In this paper by using of Suzuki-type approach we introduce the new contractive condition in the framework of non-Archimedean fuzzy metric spaces. We prove also the corresponding coincidence fixed point theorem for two mappings in this framework. Finally, two examples are presented to verify the effectiveness and applicability of our main results.

In this paper, we introduce and investigate the following subclass 1+1γzf′(z)+λz2f″(z)λzf′(z)+(1−λ)f(z)−1≺φ(z)0≤λ≤1,γ∈C∖{0}$$\begin{array}{} \displaystyle 1+\frac{1}{\gamma }\left( \frac{zf'(z)+\lambda z^{2}f''(z)}{\lambda zf'(z)+(1-\lambda )f(z)}-1\right) \prec \varphi (z)\qquad\left( 0\leq \lambda \leq 1,\gamma \in \mathbb{C} \smallsetminus \{0\}\right) \end{array} $$ of analytic functions, φ is an analytic function with positive real part in the unit disk 𝔻, satisfying φ (0) = 1, φ '(0) > 0, and φ (𝔻) is symmetric with respect to the real axis. We obtain the upper bound of the second Hankel determinant | a 2 a 4 − a32$\begin{array}{} a^2_3 \end{array} $ | for functions belonging to the this class is studied using Toeplitz determinants. The results, which are presented in this paper, would generalize those in related works of several earlier authors.